Bulk universality for generalized Wigner matrices

Abstract

Consider (N × N) Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure (\nu_{ij}) with a subexponential decay. Let (\sigma_{ij}^2) be the variance for the probability measure (\nu_{ij}) with the normalization property that (\sum_i\sigma_{ij}^2 = 1) for all j. Under essentially the only condition that (c\leq N\sigma_{ij}^2 \leq c^{−1}) for some constant (c > 0), we prove that, in the limit (N \rightarrow \infty), the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale (M^{−1}).